Logarithmic integral function

See also logarithmic integral for other senses.

In mathematics, the logarithmic integral function or integral logarithm li(x) is a non-elementary function defined for all positive real numbers ${\displaystyle x\neq 1}$ by the definite integral:

${\displaystyle {\rm {li}}(x)=\int _{0}^{x}{\frac {dt}{\ln(t)}}\;.}$

Here, ln denotes the natural logarithm. The function 1/ln (t) has a singularity at t = 1, and the integral for x > 1 has to be interpreted as a Cauchy principal value:

${\displaystyle {\rm {li}}(x)=\lim _{\varepsilon \to 0}\left(\int _{0}^{1-\varepsilon }{\frac {dt}{\ln(t)}}+\int _{1+\varepsilon }^{x}{\frac {dt}{\ln(t)}}\right)\;.}$

Sometimes instead of li the offset logarithmic integral is used, defined as ${\displaystyle {\rm {Li}}(x)={\rm {li}}(x)-{\rm {li}}(2)}$.

The growth behavior of this function for x → ∞ is

${\displaystyle {\rm {li}}(x)=\Theta \left({x \over \ln(x)}\right)\;.}$

(see big O notation).

Series representation

The function li(x) is related to the exponential integral Ei(x) via the equation

${\displaystyle {\hbox{li}}(x)={\hbox{Ei}}(\ln(x))\quad {\hbox{for all positive real }}x\neq 1.}$ 

This leads to series expansions of li(x), for instance:

${\displaystyle {\rm {li}}(e^{u})={\hbox{Ei}}(u)=\gamma +\ln u+\sum _{n=1}^{\infty }{u^{n} \over n\cdot n!}\quad {\rm {for}}\;u\neq 0\;,}$ 

where γ ≈ 0.57721 56649 01532 ... is the Euler-Mascheroni gamma constant. The function li(x) has a single positive zero; it occurs at x ≈ 1.45136 92348 ...; this number is known as the Ramanujan-Soldner constant.

Number theoretic significance

The logarithmic integral finds application in many areas, in particular it is used is in estimates of prime number densities, such as the prime number theorem:

${\displaystyle \pi (x)\sim {\hbox{li}}(x)\sim {\hbox{Li}}(x)}$ 

where π(x) denotes the number of primes smaller than or equal to x.

See also

• Jørgen Pedersen Gram

References

• Milton Abramowitz and Irene A. Stegun, eds. Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. New York: Dover, 1972. (See Chapter 5)